Definitions for Mapping stacks of topological stacks

QAlgorithms.Papers.Independent.Intakethird100.P59.CylinderPullback

For topological spaces $Q,T$ whose underlying types lie in a common universe and a continuous map $q:Q\to T$, let $I=[0,1]\subseteq\mathbb R$ with its usual subspace topology. The space $CylinderPullback(q)$ has underlying set

$$\left\{((s,t),x)\in (I\times T)\times Q:\ t=q(x)\right\}.$$

Its topology is formed by first giving $I\times T$ the ordinary product topology, then giving $(I\times T)\times Q$ the compactly generated modification of its product topology, and finally taking the compactly generated modification of the subspace topology on the displayed subset. Both endpoints $s=0$ and $s=1$ are included; $Q$ and $T$ may be empty.

QAlgorithms.Papers.Independent.Intakethird100.P59.IsRightPrincipal

Let $G$ be a group, let $G,E,B$ be topological spaces whose underlying types lie in a common universe, let $p:E\to B$ be continuous, and let $a:E\times G\to E$ satisfy $a(e,1)=e$ and $a(a(e,g),h)=a(e,gh)$. The property $IsRightPrincipal(p,a)$ means that the action map is continuous when $E\times G$ has the compactly generated modification of its product topology; that $p(a(e,g))=p(e)$ for every $e\in E$ and $g\in G$; and that every $b\in B$ belongs to an open subset $U\subseteq B$ for which there is a homeomorphism

$$\phi:p^{-1}(U)\longrightarrow U\times G.$$

Here $p^{-1}(U)$ and $U$ each carry the compactly generated modification of their respective subspace topologies, and $U\times G$ carries the compactly generated modification of the product topology formed from those topologies. For every $z\in p^{-1}(U)$, the underlying point of $B$ in the first coordinate of $\phi(z)$ must equal $p(z)$. Moreover, for every such $z$ and every $g\in G$, the point $a(z,g)$ must belong to $p^{-1}(U)$, and, regarded as a point of that subspace, must satisfy

$$\phi(a(z,g))=\bigl(\phi(z)_1,\phi(z)_2g\bigr).$$

The neighborhood requirement ranges over all points of $B$ and is vacuous if $B$ is empty.

QAlgorithms.Papers.Independent.Intakethird100.P59.KProduct

For arbitrary types $X$ and $Y$, possibly in different universes, $KProduct(X,Y)$ is the Cartesian product $X\times Y$: its elements are ordered pairs $(x,y)$ with $x\in X$ and $y\in Y$. This definition specifies only the underlying type. Either type may be empty.

QAlgorithms.Papers.Independent.Intakethird100.P59.KSubspace

For any type $X$ and subset $s\subseteq X$, $KSubspace(X,s)$ is the subtype consisting of elements $x\in X$ together with the condition $x\in s$. This definition specifies only the underlying type; in particular, it is empty when $s$ is empty.

QAlgorithms.Papers.Independent.Intakethird100.P59.RightAction

For a group $G$ and an arbitrary type $E$, possibly in different universes, a right action consists of a function $a:E\times G\to E$ satisfying

$$a(e,1)=e,\qquad a(a(e,g),h)=a(e,gh)$$

for every $e\in E$ and every $g,h\in G$, where $1$ is the identity of $G$. No topology or continuity is part of this structure, and $E$ may be empty.

QAlgorithms.Papers.Independent.Intakethird100.P59.basedTwistedLoopSet

For any group $G$ equipped with a topology and any $\alpha\in G$, this defines the subset of continuous maps $\gamma:\mathbb R\to G$ satisfying both

$$\gamma(t+1)=\alpha\,\gamma(t)\,\alpha^{-1} \qquad\text{for every }t\in\mathbb R, \qquad\text{and}\qquad \gamma(0)=1,$$

where $1$ is the identity of $G$. All real times are included, and continuity of the group operations is not assumed.

QAlgorithms.Papers.Independent.Intakethird100.P59.identityBasedTwistedLoopSet

For any group $G$ equipped with a topology and any $\alpha\in G$, let $G_1$ be the path component of the identity $1$: the subset of points joined to $1$ by a continuous path, equipped with its ordinary subspace topology. This defines the subset of continuous maps $\gamma:\mathbb R\to G_1$ whose underlying $G$-valued map $\bar\gamma$ satisfies

$$\bar\gamma(t+1)=\alpha\,\bar\gamma(t)\,\alpha^{-1} \qquad\text{for every }t\in\mathbb R, \qquad\text{and}\qquad \bar\gamma(0)=1.$$

Thus every value of $\gamma$ is required to lie in the path component of $1$, and the equations are equations in $G$. No assumption requires $\alpha$ itself to lie in that component, and continuity of the group operations is not assumed.

QAlgorithms.Papers.Independent.Intakethird100.P59.includeIdentityLoop

For any group $G$ equipped with a topology, and any continuous map $\gamma:\mathbb R\to G_1$, where $G_1$ is the path component of the identity $1\in G$ with its subspace topology, $\operatorname{includeIdentityLoop}(\gamma)$ is the continuous map $\mathbb R\to G$ given by $t\mapsto\gamma(t)$, viewed as a point of $G$. Thus this operation simply composes $\gamma$ with the inclusion $G_1\hookrightarrow G$. No condition that $\gamma(0)=1$, periodicity condition, or compatibility between the group operations and the topology is assumed.

QAlgorithms.Papers.Independent.Intakethird100.P59.instance_17

For arbitrary topological spaces $X$ and $Y$, this instance equips the underlying Cartesian product $X\times Y$ with the compactly generated modification of its ordinary product topology. The compactly generated modification is taken in the larger of the universes containing $X$ and $Y$. No separation or nonemptiness assumptions are imposed on either space.

QAlgorithms.Papers.Independent.Intakethird100.P59.instance_25

For any topological space $X$ and any subset $s\subseteq X$, this instance equips the type of points of $s$ with the compactly generated modification of the ordinary subspace topology inherited from $X$, with the modification taken in the universe containing $X$. The subset need not be open, closed, or nonempty.

QAlgorithms.Papers.Independent.Intakethird100.P59.singletonRightAction

On the one-element type $PUnit=\{\ast\}$, equipped with its one-element group structure, this defines a right action on $PUnit$ by $a(\ast,\ast)=\ast$. It satisfies $a(e,1)=e$ and $a(a(e,g),h)=a(e,gh)$ for all elements $e,g,h$ of the singleton.

QAlgorithms.Papers.Independent.Intakethird100.P59.twistedLoopSet

For any group $G$ equipped with a topology and any $\alpha\in G$, this defines the subset of continuous maps $\gamma:\mathbb R\to G$ consisting exactly of those satisfying

$$\gamma(t+1)=\alpha\,\gamma(t)\,\alpha^{-1} \qquad\text{for every }t\in\mathbb R.$$

The condition ranges over all real times, including negative times; no value at $0$ is prescribed, and continuity of the group operations is not assumed.